An L-sentence that "fixes" a model. I'm working through David Marker's Model Theory, and I'm stuck on Exercise 1.4.2 b. It states

Let $\mathcal L$ be any finite language and let $\mathcal M$ be a finite $\mathcal L$-structure. Show
  that there is an $\mathcal L$-sentence $\phi$ such that $\mathcal N \vDash \phi$ if and only if $\mathcal N$ is isomorphic to $\mathcal M$.

I think I understand the idea here, we want to basically reproduce the interpretation of each $n$-ary function $f \in \mathcal F(\mathcal L)$ and each $n$-ary relation $R \in \mathcal R(\mathcal L)$, and the cardinality of $\mathcal M$ into an $\mathcal L$-sentence.
My main issue is that I don't know how to refer to arbitrary elements in $\mathcal M$ that aren't symbols in $\mathcal L$. For example, in the language of fields, in the structure $(\mathbb R, +, -, \cdot, \div, 0, 1)$, I don't know how to refer to the number $\pi$, for example. Even if the structure is finite, I don't see how to refer to arbitrary elements in the underlying set.
Any advice would be appreciated.
 A: The idea is to use quantification to require the cardinality of the satisfying model to be $|\mathcal M|$, and to require that the functions, relations, and constants behave the same as in $\mathcal M$. So something like:
$$\phi = \exists m_1, \dots, m_k \forall m_0 \left(\text{$m_0$ must be one of the $m_i$'s}\right)\land \\\left( \bigwedge_{f \in \mathcal F} \text{describe the function $f$} \right) \land \\\left( \bigwedge_{r \in \mathcal R} \text{describe the relation $r$} \right) \land \\\left( \bigwedge_{c \in \mathcal C} \text{describe the constant $c$} \right) \quad$$
Where $k = |\mathcal M|$. These four "components" of the formula describe


*

*That the cardinality of a satisfying model is $|\mathcal M|$.

*That the interpretation of each function is the same as in $\mathcal M$.

*That the interpretation of each relation is the same as in $\mathcal M$.

*That the interpretation of each constant is the same as in $\mathcal M$.


Any structure that satisfies these four conditions is isomorphic to $\mathcal M$.
